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A Generalization of Singer's Theorem: Any Finite Pappian A-Plane Is PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 71, Number 2, September 1978 A GENERALIZATIONOF SINGER'S THEOREM MARK P. HALE, JR. AND DIETER JUNGN1CKEL1 Abstract. Finite pappian Klingenberg planes and finite desarguesian uniform Hjelmslev planes admit abelian collineation groups acting regularly on the point and line sets; this generalizes Singer's theorem on finite desarguesian projective planes. Introduction. Projective Klingenberg and Hjelmslev planes (briefly K- planes, resp., //-planes) are generalizations of ordinary projective planes. The more general A-plane is an incidence structure n whose point and line sets are partitioned into "neighbor classes" such that nonneighbor points (lines) have exactly one line (point) in common and such that the "gross structure" formed by the neighbor classes is an ordinary projective plane IT'. An //-plane is a A-plane whose points are neighbors if and only if they are joined by at least two lines, and dually. A finite A-plane possesses two integer invariants (/, r), where r is the order of IT and t2 the cardinality of any neighbor class of n (cf. [3], [6]). Examples of A-planes are the "desarguesian" A-planes which are constructed by using homogeneous coordinates over a local ring (similar to the construction of the desarguesian projective planes from skewfields); for the precise definition, see Lemma 2. If R is a finite local ring with maximal ideal M, one obtains a (/, r)-K-p\ane with t = \M\ and r = \R/M\. The A-plane will be an //-plane if and only if A is a Hjelmslev ring (see [7] and [»])• We recall that a projective plane is called cyclic if it admits a cyclic collineation group which acts regularly on the point and line sets. Singer's theorem [10] asserts that every finite desarguesian projective plane is cyclic. The notion of a cyclic projective plane was generalized by Jungnickel to that of a "regular" A-plane: the main requirement is the existence of an abelian collineation group acting regularly on the point and line sets (see Definition 2). In [4], a particular class of commutative //-rings was shown to give rise to regular //-planes by a computational argument. In the present paper, we prove a generalization of Singer's theorem: Any finite pappian A-plane is Received by the editors June 1, 1977. AMS (MOS) subject classifications(1970). Primary 50D35, 05B25; Secondary 05B10. Key words and phrases. Hjelmslev plane, Klingenberg plane, Singer's theorem, regular collineation group. 'The second author acknowledges the hospitality of the University of Florida during the time of his research. 280 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A GENERALIZATION OF SINGER'S THEOREM 281 regular. As regarding nonpappian planes, we show thay any finite desar- guesian //-plane with t = r is regular. We will only consider finite incidence structures. All rings considered are associative with unit. The following lemma is well known (see e.g. [1, Chapitre 2]). Lemma I. In a local ring R, the (unique) maximal ideal M consists precisely of the nonunits of R. If R is finite, M is nilideal. Definition 1 (Klingenberg). A local ring R is called a Hjelmslev ring if the maximal ideal M consists precisely of the zero divisors of R and if for all a, b £ M (i) a £ bM or b E aM and (ii) a E Mb or b E Ma. Lemma 2 (Klingenberg). Let R be a finite local ring with maximal ideal M, R* = R\M be the set of units of R. Define an incidence structure Tl(R) as follows: points are the homogeneous triples R*p = R*(Pq,Px,P2) where not all p¡ are in M; lines are the homogeneous triples uTR* = (uQ, ux, u2)TR* where not all u¡ are in At"; incidence is defined by R*pIuTR* if and only ifpuT = 0. Then U(R) is a (t, r)-K-plane with t = \M\ and r = \R/M\. U(R) is an H-plane if and only if R is a Hjelmslev ring. Any nonsingular 3 X 3-matrix A over R induces a collineation of H(R ) by putting R*pH>R*(pA~x) and uTR*\-+(\uT)R*. For a proof, see [7] and [8]. Definition 2. A AT-planen is called desarguesian if it can be represented as a K-plane Ti(R) over a local ring R, as in Lemma 2. If R is commutative, n is called pappian. Definition 3 (Jungnickel [4]). A AT-planeII is called regular if it admits an abelian collineation group G = Z © K satisfying (i) G acts regularly on the point and line sets of n; (ii) K acts regularly on each neighbor class (of points or lines) of n. One notes that (ii) is equivalent to (ii') Z acts regularly on the point and line sets of IT. Proposition. Let R be a finite commutative local ring with maximal ideal M. Then there is a ring structure definable onS = R®R®Rin which S is a commutative local ring extending R and having maximal ideal N = M © M © M. Proof. R/M is a finite field F. Choose a monk cubic polynomial f(x) over F, and let f(x) be any monic cubic preimage of f(x) with respect to the natural map R[x] -^ F[x\ We will show that S '■—R[x]/(f) has the desired License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 282 M. P. HALE, JR. AND DIETER JUNGNICKEL properties. Certainly S is a commutative extension of R, which appears in S as the image of the constant polynomials. As an Ä-module, S = R © R © R since /is cubic. Now consider the ideal A = (M) = SM. Then S/N = R[x]/ (f)/{M[x] + (/))/ (/) a R[x]/ (M[x] + (/)) s R[x]/M[x]/(M[x] + (f))/M[x] = F[x]/(f). By the construction off, this last quotient is a 3-dimensional field extension F of F. Thus A is maximal in S. Since M is a nilideal in R, and S is commutative, the ideal A = SM is also a nilideal and therefore lies in every maximal ideal of S. This shows that A is the unique maximal ideal of S, and S is local. Theorem 1. A finite pappian K-plane H is regular. Proof. Let n be coordinated over the commutative local ring R, say \R\ = qn+x, \N\ = q", M the maximal ideal of R. (So II is a (q", q)-A-plane.) Let S be a 3-dimensional extension of R, as constructed in the Proposition. Let a G S* '■=S\N (the group of units of S). Then the mapping d»0: R3 —>R3 (i.e., <ba:S —>S) with <pa:xi->xa is a bijective linear mapping of R3. Hence S* is a group of linear transformations of R 3. But n is coordinatized by homogeneous triples R*(Po,Px,p2), resp., (u0, «,, u2)TR* of elements of R, i.e., n is coordinatized by the elements of S*/R*. Thus G '■=S*/R* acts as a transitive abelian collineation group of n. For reasons of cardinality, G must be regular on n: G has (q3n+3 — qiny^n+x _ qn^ = q2n^2 + q + ,) eiementSj wftich agrees with the number of points (resp. lines) of n (cf. [3] and [6]). As G is abelian and (q2n, q2 + q + 1) = 1, G splits into G = Z © A, where Z has order q2 + g + 1 and A has order q2". Consider the homomorphism from S to F. This induces a homomorphism from n (as coordinatized by S*/R*) to n' (as coordinatized by F*/F*). The image of S* induces a Singer cycle in n', i.e., a cyclic group of order q2 + q + 1; by reasons of cardinality, this is the image of Z; so II is regular with respect to G. Theorem 1 yields together with [4, Corollary 2.8] a more elegant proof of the following result, which is crucial for the recursive construction of regular //-planes (cf. [4, §5] and [5]). Corollary (Jungnickel [4, Corollary 4.3]). There exist (qn, q)-H-matrices for all prime powers q and all natural numbers n. Definition 3 (Lüneburg [9]). A (t, r)-//-plane n is called uniform if t = r. Theorem 2. A finite uniform desarguesian H-plane n is regular. Proof, n is represented over a "uniform" Hjelmslev ring R (i.e., M2 = 0; see [2]). If R is in fact commutative, the assertion follows by Theorem 1. Otherwise, Theorem 6 of [2] asserts that R has the following structure: Let License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A GENERALIZATION OF SINGER'S THEOREM 283 F = GF(<?):= R/M. R is defined on F X F by componentwise addition and multiplication (a, b)(c, d) '■= (ac, ad + be"), where a is a nontrivial automorphism of F. (These //-rings are due to Kleinfeld [6].) Consider the polynomial p(x) '■=x3 — x over F, p is not one-to-one (as p(0) = p(\) = 0), hence not onto; let u be chosen such that x3 — x — u has no root and is therefore irreducible. This defines, by adjoining a root a, the 3-dimensional extension field GF(<73).Multiplication by a (x\->xa) induces a linear transformation on F3; its matrix is 0 0 1 5 = 1 0 u 0 1 0 As in the proof of Singer's theorem one shows that S induces a regular collineation group Z on the projective plane n' over F; as F is a subring of R, it is easy to see that Z in fact acts semiregularly on n.
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